106 NEWTON S PRINCIPIA. 



the periodic time of M moving in a similar figure round 

 E at rest, T : t :: \/E : &amp;gt;/M + E. Further if these bo 

 dies move with forces inversely as the squares of the 

 distances round their centres of gravity, and A be the 

 greater axis of the ellipse described by M round E, a 

 the greater axis of the ellipse it would describe round 

 E at rest in the same time, and if M + E : m : : n : E, 

 then A : a : : M + E : m. Hence, if we have the periodic 

 times of the planets, we can find the greater axes of their 

 orbits by taking A 3 to 3 in the proportion of T 2 to t 2 

 (the ellipse being supposed described round the sun), and 



multiplying it by - So the masses may, likewise, be 



found from the distances. 



The motions and paths of bodies thus mutually acting 

 are now to be considered. And first our author shows, 

 that if two bodies act on each other, and move without any 

 other, or foreign, influence whatever, their motion will 

 be the same as if, instead of acting on one another, some 

 third body placed in their centre of gravity acted upon 

 each of them with the same force with which each acts 

 on the other ; and the same law will prevail (but referred 

 to the distances from the centre) which prevailed in their 

 mutual actions when referred to their distances from 

 each other. Suppose the bodies M and E to attract 

 with forces directly as their masses M and E, and in 

 versely as any power n of their distances, that is, suppose 



their attraction to be as ^ n , and =p and that the dis 

 tances of the centre from M and E are C and c re 

 spectively ; then because C : c : : E : M, and C : C + c 

 (or D) :: E : E + M, a body in the centre will attract 



M with a force as pp if it be equal to .., - , that is 



